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Non-injectivity of the $$q$$-deformed von Neumann algebra. (English) Zbl 1060.46051
Summary: We prove that the von Neumann algebra generated by $$q$$-Gaussians is not injective as soon as the dimension of the underlying Hilbert space is greater than 1. Our approach is based on a suitable vector valued Khintchine type inequality for Wick products. The same proof also works for the more general setting of a Yang-Baxter deformation. Our techniques can also be extended to the so called $$q$$-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter case, we obtain the non-injectivity under some assumption on the spectral set of the positive operator associated with the deformation.

##### MSC:
 46L65 Quantizations, deformations for selfadjoint operator algebras 46L54 Free probability and free operator algebras
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